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A prior estimate and existence of positive solutions of semilinear elliptic equation with the third boundary value problem. (English) Zbl 1136.35379

This paper deals with a priori estimate and existence of positive solution of the following equations
\[ \begin{gathered} Lu= f(u)\quad\text{in }\Omega,\\ a_{ij}{\partial u\over\partial x_i}\cos(n, x_j)+\alpha u= 0\quad\text{on }\partial\Omega,\end{gathered} \]
where operator
\[ L= -a_{ij}(x){\partial^2\over\partial x_i\partial x_j}+ a_i(x){\partial\over\partial x_i}+ a(x) \]
and nonlinear term has the form \(f(u)= u^p,\;p\in[1,{N+2\over N-2})\). Here \(\Omega\) is a bounded domain in \(\mathbb{R}^N\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B45 A priori estimates in context of PDEs
35J60 Nonlinear elliptic equations
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